Questions tagged [proof-techniques]

Questions about general methods and techniques for proving multiple theorems. When asking about the proof of a single statement, use tags relating to what the proof is about instead.

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If string x is conjugate to string y, how can you prove that the reverse of x is conjugate to the reverse of y?

I am relatively new to writing proofs and I am stuck trying to prove that if $x$ ~ $y$, then $x^R$ ~ $y^R$. Any help would be appreciated!
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1answer
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Prove there is an algorithm that needs at most n-1 comparisons to check if n-element array has all equal elements?

Question: Prove that there exists an algorithm that can decide using at most n-1 comparisons whether a n-element array contains only equal numbers. We use an algorithm that loops through all the ...
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1answer
41 views

Subtraction on Big Theta notation

This is a question I got for an assignment, and I have been stuck on it for the past few days. Prove that $\Theta(n)+\Theta(n-1) = \Theta(n)$ Does it follow that $\Theta(n) = \Theta(n)-\Theta(n-1)$ I ...
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1answer
241 views

Proof via induction for small-step semantics

I'm doing a course in Computer Programming Languages and I'm trying to prove the following (roughly following Pierce's Types and Programming Languages book): if $t \rightarrow^* t'$ then $if\; t\; ...
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3answers
260 views

Prove (p → ¬q) is equivalent to ¬(p ∧ q)

I need to prove the above sequent using natural deduction. I did the first half already i.e. I proved $(p\rightarrow\neg q)\rightarrow \neg (p \wedge q)$, but I'm stuck on where to start for the ...
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How to prove that some function $f$ is {time,space}-constructible function

Is there a standard method to prove or disprove that some functions are or aren't time or space constructible? can you give me a way to check them ? or an example ?
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How to prove that a language is context-free?

There are many techniques to prove that a language is not context-free, but how do I prove that a language is context-free? What techniques are there to prove this? Obviously, one way is to exhibit ...
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1answer
337 views

reduction from ALLTM to ETM

I am trying to understand why this "proof" of ETM undecidability is wrong. ALLTM={ < M >|M is a TM, L(M)=∑*} ETM={< M >|M is a TM, L(M)=∅} We know that ALLTM is undecidable, lets ...
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Mutual Friends in a Network?

I always seem to have trouble finding a formal way to analyze this (be through proofs or whatever). The problem statement is as such: If A and B are friends, and B and C are friends, then A and C are ...
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Advantages of amortized analysis

I understood what amortized analysis does, but can anyone tell me what is the main purpose of this kind of analysis? What I understood: Let say we have 3 three operations a,b,c used 1,2 and 3 times ...
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100 views

Proof for time complexity of Insertion (k-proximate) Sort equals O(nk)

The following is the definition for Proximate Sorting given in my paper: An array of distinct integers is k-proximate if every integer of the array is at most k places away from its place in the array ...
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Suppose f(n) = O(h(n)) and g(n) = O(h(n)). Is f(n) * g(n) = O(h(n) * h(n))?

I understand this should be a relatively easy proof, but I can't seem to understand how to do it. I know that, by Big O definition: there exists some value $c_1$ where $f(n) \le c_1 \cdot h(n)$ for ...
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What is the contradiction of statement "if $n^2$ is odd then n is odd"

In my opinion the contradiction should be-: If $n^2$ is even then n is even. But it is written in my discrete mathematics book that, "n is even then $n^2$ is odd". How do we find ...
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1answer
24 views

What do we conclude from diagonalization principle?

I understand $R_{fa}$ etc. I understand why the diagonals are higlighted. I understand D={a,d,f}. But I don't understand what is the conclusion we derive from this?
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Show that any monotone Boolean function is computable by a circuit containing only AND and OR gates

A Boolean function $f : \{0, 1\}^n → \{0, 1\}$ is called monotone if changing any of the $n$ input bits $x_1, \ldots , x_n$ from $0$ to $1$ can only ever change the output $f(x_1, \ldots ,x_n)$ from $...
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3answers
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Algorithm for solving a mixed integer programming problem in polynomial time?

I have the following mixed integer programming (MIP) problem: $$ \begin{array}{rll} \text{Maximize } & z=k \\ \text{subject to } & a_ik - m_i \geq 0 & (i=1,\dots,n) \\ & b_ik - m_i \...
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1answer
21 views

Karatsube-Ofman runtime complexity computation

I have a question and didn't understand the solution, since we didn't take how to do it in the lecture and it's not explained in the solution sample. Question: One can generalize the Karatsube-Ofman ...
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2answers
175 views

Big theta notation in substitution proofs for recurrences

Often in CLRS, when proving recurrences via substitution, $\Theta(f(n))$ is replaced with $cf(n)$. For example, on page 91, the recurrence $$ T(n) = 3T(⌊n/4⌋) + \Theta(n^2) $$ is written like so in ...
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How to prove a problem is NOT NP-Complete?

Is there any general technique for proving a problem NOT being NP-Complete? I got this question on the exam that asked me to show whether some problem (see below) is NP-Complete. I could not think of ...
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1answer
124 views

How to verify-: A language over Σ is also a language over any alphabet that is a superset of Σ?

Context-: A language over Σ need not include strings with all symbols of Σ Thus, a language over Σ is also a language over any alphabet that is a superset of Σ. https://www.univ-orleans.fr/lifo/...
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Red-Black tree height from CLRS

The lemma 13.1 of CLRS proves that the height of a red black tree with $n$ nodes is $$h(n) \leq 2\log_2(n+1)$$ There's a subtle step I don't understand. The property 4 reported at the beginning of ...
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71 views

Validity of proof by contradiction

I had a doubt in the proof by contradiction technique. Under this technique, we assume the negation of what we want to prove as true, then show that assuming so generates a contradiction. Since a ...
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1answer
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Is it common to prove that some code is the simplest way to achieve something?

I have a simple program which achieves a certain functionality. I’m interested to know if it can be proven that the steps in the program are the theoretically simplest way to achieve those results. Is ...
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Uniqueness of solution in Arden's theorem

Geeksforgeeks contains a proof of Arden's theorem, asserting that $R=QP^*$ is the unique solution to $R=Q+RP$. The proof is reproduces below. My question is: What is the logical reasoning to prove ...
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NP-hardness proof of an optimization problem with real values and rational input in the decision problem

I'm studying complexity theory and I have the below question regarding $NP$-hardness proofs of optimization problems with real values. Any reference is much appreciated. For the question, take the ...
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35 views

NP-hardness proof of an optimization problem with real values and real input in the decision problem

Question - Let's suppose we have an optimization problem $\mathcal{P}$ with a real-valued measure function and the decision version of the optimization problem $\mathcal{P}_D$ (please see definitions ...
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1answer
122 views

Invariant vs Assertion vs lemma

I am reading Distributed Algorithms by Nancy Lynch. I have come across lemmas, assertions and invariants, but I do not understand the difference between them. I think lemma means an intermediate ...
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Are there any specific problems known to be undecidable for reasons other than diagonalization, self-reference, or reducibility?

Every undecidable problem that I know of falls into one of the following categories: Problems that are undecidable because of diagonalization (indirect self-reference). These problems, like the ...
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1answer
39 views

How can I use induction for proving termination of a string rewriting system?

If we have a string rewriting system within the alphabet $\{X,Y\}^*$ and the rule $XY\to YX$. How can we prove by induction that on every string input the system terminates?
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1answer
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M does not accept [M] | 'Correction' of proof possible?

The language $D=\{[M]|M([M])=0\}$ is not decidable because of the following argument: Suppose there was a $TM \space M_D$ that decides $D$. Then if we gave $M_D \space [M] $, there would be two ...
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How to prove greedy algorithm is correct

I have a greedy algorithm that I suspect might be correct, but I'm not sure. How do I check whether it is correct? What are the techniques to use for proving a greedy algorithm correct? Are there ...
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1answer
146 views

Proof that for every $k > 1$, there exists a language $A_k \subseteq \{0, 1\}^*$ s.t. a DFA accepting $A_k$ has $k$ states but no less

I am trying to prove that for every $k > 1$, there exists a language $A_k \subseteq \{0, 1\}^*$ such that a DFA accepting $A_k$ has $k$ states but no less. I thought about proving this in two ways: ...
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1answer
55 views

How to prove NP-hardness of a Hamiltonian Path problem by reducing longest-path problem?

I know how to prove longest-path problem by reducing Hamiltonian Path problem. Here I want to prove NP-hardness of a Hamiltonion Path problem by reducing longest-path problem. (pretend we know longest-...
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56 views

Please help me understand this proof of the undecidability of "Do two halting Turing machines accept the same language?"

Do two halting Turing machines accept the same language? Proof that it is undecidable(credit to another user on this website: "Tom van der Zanden"): Let M be an arbitrary Turing machine. Let ...
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Proving that $f(n) \not\in O(n)$ given that $f(n) \in \Theta(n^2)$ and the formal definitions of Big-Oh and Theta

So far I've understood that because of the definition of $\Theta$, we have $c_1n^2 \le f(n) \le c_2n^2$. I'm not sure how to proceed from there.
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2answers
171 views

Finding an algorithm to return the $\log n$ largest element in an array

I have just proved that for every $\alpha, \beta>0 : (\log n)^\alpha=O(n^\beta)$. Now, given an array of $n$ elements, I want to find an efficient comparison based algorithm for finding the $\log n$...
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1answer
114 views

How can I prove the following problem is NP complete?

The problem: I have a list $\displaystyle S=\{s_{1} ,s_{2} ,\dotsc ,s_{n}\}$ places. Each unordered pair of places has cost and gain: $\displaystyle c\{s_{i} ,s_{j}\} \in \mathbb{N}$, $\displaystyle g\...
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2answers
794 views

Longest path length in an undirected tree, can we prove this algorithm is correct (which it is)?

Hello I solved this leetcode https://leetcode.com/problems/tree-diameter/ question reserved for people who pay the subscription. The question: Given an undirected tree (tree is not disjoint), ...
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Proving Minimum Sequence Disjoint Set

Prove that 9 is the minimum number of calls to make-set, union-set, find-set such that a disjoint set union using weight (number of nodes) by path compression and disjoint set union using rank (upper ...
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1answer
96 views

Difficulty in understanding the proof of "Every context-sensitive language L is recursive" as given in the Peter Linz text

I was going through the automata text by Peter Linz. There I came across the proof the theorem below. I could not quite get the portion of the proof in bolds. Every context-sensitive language L is ...
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How to use butterfly network for data copy?

I know butterfly networks (and benes as well) allow routing a packet from any input to any output node. Congestion is $\sqrt{n}$ but with bene it can be $1$. Now assume that in a butterfly network, ...
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1answer
155 views

Language of CFG: $S \to aS | aSbS | \varepsilon$

I'm trying to prove that the language L generated by the CFG $S \to aS | aSbS | \varepsilon$ is the language $L=\{ w \in \{a,b\}^*: \text{every prefix of $w$ has at least as many $a$'s as $b$'s} \}$.I ...
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1answer
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Given CFG generates all words with equally many 0s and 1s

Here is an exercise from an introduction to computation class: Show that the following context-free grammar $G$ generates the language $L$ of words over $\{0,1\}$ with an equal number of $0$s and $1$...
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1answer
50 views

Towers of Hanoi with sufficiently many stacks, show that $T_k(n)=\Theta(n)$ for all $k\geq 2 + \frac{n-1}{2}$

I'm trying to show that for the following Towers of Hanoi general algorithm that $T_k(n)=\Theta(n)$ for all $k\geq 2 + \frac{n-1}{2}$, I'm not sure how to incorporate the restriction on $k$ into my ...
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1answer
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Inductive sequence of words in a biprefix code

Let $X = X_1 \cup X_2$ a code on an alphabet $A$, with $X_1$ a biprefix code and $X_2$ a uniform code, with $m(X_1) < m(X_2)$, i.e. the maximal length of the first is strictly lower than the second....
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1answer
87 views

Pumping Lemma Proof (Type of wcw language)

I have the language $L = \{ dkd\space \mid d \in \{a,b\}^*, k \in \{a,b\} \}$ and i have to show that it's non-regular using the pumping lemma. The structure of the language i think can be explained ...
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Can I simplify log(n+1) before showing that it is in O(log n)?

Had a question about the following: $$\log (n+1) \in O(\log n)$$ Can the left side be simplified any further or do I need to just go ahead and find a c and n that hold?
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Proving using a SAT solver that KB entails D

Suppose you're given this KB: $$KB = (A, A ⇒ B, A ⇒ C, B ∧ C ⇒ D)$$ How would you show using a SAT solver that $KB \vDash D$?
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984 views

Rigorous proof for validity of assumption $n=b^k$ when using the Master theorem

The Master theorem is a beautiful tool for solving certain kinds of recurrences. However, we often gloss over an integral part when applying it. For example, during the analysis of Mergesort we ...

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